For stratified spaces

A stratified space has a ‘fundamental n-category with duals’, which generalizes the fundamental n-groupoid of a plain old space. When a path crosses a codimension-11 stratum, “something interesting happens” – i.e., a catastrophe. So, we say such a path gives a noninvertible morphism. The idea is that going along such a path and then going back is not “the same” as having stayed put. So, going back along such a path is not its inverse, just its dual.